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Chapter 9: Problem 1

In Exercises 1 to \(6,\) each situation calls for a significance test. State the appropriate null hypothesis \(\mathrm{H}_{0}\) and alternative hypothesis \(H_{a}\) in each case. Be sure to define your parameter each time. Attitudes The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures students' attitudes toward school and study habits. Scores range from 0 to \(200 .\) The mean score for U.S. college students is about \(115 .\) A teacher suspects that older students have better attitudes toward school. She gives the SSHA to an SRS of 45 of the over 1000 students at her college who are at least 30 years of age.

Short Answer

Expert verified
\( H_{0}: \mu = 115 \); \( H_{a}: \mu > 115 \).

Step by step solution

01

Identify the Parameter

In this situation, we need to define the parameter, which represents the target value in the population. Here, the parameter is the mean SSHA score of students at least 30 years old at the college, denoted by \( \mu \).
02

Define the Null Hypothesis (H0)

The null hypothesis represents a statement of no effect or no difference. Since the mean SSHA score for U.S. college students is about 115, the null hypothesis would state that the mean score of older students is the same as this value. Therefore, the null hypothesis is \( H_{0}: \mu = 115 \).
03

Define the Alternative Hypothesis (Ha)

The alternative hypothesis represents a statement that contradicts the null hypothesis. The teacher suspects that older students have better attitudes, which would manifest as higher mean scores. Therefore, the alternative hypothesis is \( H_{a}: \mu > 115 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis
The null hypothesis, often denoted as \( H_0 \), is like an initial assumption or a baseline statement. It suggests that there is no significant change or effect, and in this context, it means that any observed differences are due to random chance. It's important in statistical testing because it sets a ground rule from which we can start. To understand it better, think about it as a kind of default assumption that will stand unless there is enough convincing evidence to suggest otherwise. For example:
  • If the null hypothesis says the mean score of older students is \( 115 \), it suggests their scores are just like the average U.S. college student.
  • If your test results show a score significantly higher or lower than \( 115 \), then we need to look at the possibility of some other factors at play.
In experiments and studies, we usually aim to disprove the null hypothesis if we suspect something different or more significant is happening.
Alternative Hypothesis
The alternative hypothesis, marked as \( H_a \), stands in direct opposition to the null hypothesis. It's a bold claim that there is a real effect or a notable difference. If we gather enough evidence, we can "reject" the null hypothesis in favor of this alternative. Using our example from the exercise:
  • The alternative hypothesis \( H_a: \mu > 115 \) suggests that older students tend to have higher SSHA scores, indicating better attitudes toward school.
  • This is more than just conjecture; it's a hypothesis that we seek to support with data.
In conducting a significance test, if the observed data substantially supports the alternative hypothesis, we might consider it valid, leading to a potential shift in understanding or decision-making.
Statistical Parameter
A statistical parameter is a numeric characteristic of a population, such as a mean or standard deviation. It's critical because it describes aspects of the population that we are interested in studying. Usually, we denote these with Greek letters like \( \mu \) for the mean or \( \sigma \) for the standard deviation.In the context of the exercise, the focus is on the mean SSHA score, denoted by \( \mu \):
  • It represents the average score of students who are at least 30 years old.
  • The parameter helps to establish and test hypotheses about the population from which our sample is drawn.
By analyzing these parameters, researchers can infer information about the entire population from just a sample. When dealing with different groups within a population, defining the right parameter is vital to ensure the accuracy and reliability of our conclusions.

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Most popular questions from this chapter

When asked to explain the meaning of the \(P\) -value in Exercise 13 , a student says, "This means there is about a \(22 \%\) chance that the null hypothesis is true." Explain why the student's explanation is wrong.

A drug manufacturer claims that fewer than \(10 \%\) of patients who take its new drug for treating Alzheimer's disease will experience nausea. To test this claim, a significance test is carried out of $$ \begin{array}{l} H_{0}: p=0.10 \\ H_{a}: p<0.10 \end{array} $$ You learn that the power of this test at the \(5 \%\) significance level against the alternative \(p=0.08\) is 0.29 . (a) Explain in simple language what "power \(=0.29 "\) means in this setting. (b) You could get higher power against the same alternative with the same \(\alpha\) by changing the number of measurements you make. Should you make more measurements or fewer to increase power? Explain. (c) If you decide to use \(\alpha=0.01\) in place of \(\alpha=0.05\), with no other changes in the test, will the power increase or decrease? Justify your answer. (d) If you shift your interest to the alternative \(p=0.07\) with no other changes, will the power increase or decrease? Justify your answer.

In a recent year, \(73 \%\) of firstyear college students responding to a national survey identified "being very well-off financially" as an important personal goal. A state university finds that 132 of an SRS of 200 of its first-year students say that this goal is important. Is there convincing evidence at the \(\alpha=0.05\) significance level that the proportion of all first-year students at this university who think being very well-off is important differs from the national value, \(73 \% ?\)

Exercises 21 and 22 refer to the following setting. Slow response times by paramedics, firefighters, and policemen can have serious consequences for accident victims. In the case of life-threatening injuries, victims generally need medical attention within 8 minutes of the accident. Several cities have begun to monitor emergency response times. In one such city, the mean response time to all accidents involving life-threatening injuries last year was \(\mu=6.7\) minutes. Emergency personnel arrived within 8 minutes on \(78 \%\) of all calls involving life-threatening injuries last year. The city manager shares this information and encourages these first responders to "do better." At the end of the year, the city manager selects an SRS of 400 calls involving life-threatening injuries and examines the response times. (a) State hypotheses for a significance test to determine whether the average response time has decreased. Be sure to define the parameter of interest. (b) Describe a Type I error and a Type II error in this setting, and explain the consequences of each. (c) Which is more serious in this setting: a Type I error or a Type II error? Justify your answer.

Bottles of a popular cola are supposed to contain 300 milliliters (ml) of cola. There is some variation from bottle to bottle because the filling machinery is not perfectly precise. An inspector measures the contents of six randomly selected bottles from a single day's production. The results are $$ \begin{array}{llllll} 299.4 & 297.7 & 301.0 & 298.9 & 300.2 & 297.0 \end{array} $$ Do these data provide convincing evidence that the mean amount of cola in all the bottles filled that day differs from the target value of \(300 \mathrm{ml} ?\)
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